Proof.Suppose that \(\nu \)  is such that \(\theta ^\nu \ge 9 \beta ^2/\alpha ^2\)  and \(\Vert p^\nu -p\Vert _\infty \le \alpha /3\) . Let \(\tilde{f}^\nu (u,x) = f^\nu (u,x) - \langle y^\nu , u\rangle \) . The two functions \(f\)  and \(\tilde{f}^\nu \)  are lsc. This allows us to slightly refine [1]} (details omitted) and confirm that
\(\operatorname{dist}_*\big ((u^\nu ,x^\nu ), ~\delta \mbox{-}\hspace{-1.70709pt}\mathop {\rm argmin}\nolimits _{u,x} f(u,x)\big ) \le  d \hat{\hspace{-1.49994pt}l }_\rho (\operatorname{epi}f, \operatorname{epi}\tilde{f}^\nu )\) 
